Two-dimensional generalization of the Muller root-finding algorithm and its applications
arXiv:1005.5375
Abstract
We propose a new algorithm for solving a system of two nonlinear transcendental equations with two complex variables based on the Muller algorithm. The two-dimensional Muller algorithm is tested on systems of different type and is found to work comparably to Newton's method and Broyden's method in many cases. The new algorithm is particularly useful in systems featuring the Heun functions whose complexity may make the already known algorithms not efficient enough or not working at all. In those specific cases, the new algorithm gives distinctly better results than the other two methods. As an example for its application in physics, the new algorithm was used to find the quasi-normal modes (QNM) of Schwarzschild black hole described by the Regge-Wheeler equation. The numerical results obtained by our method are compared with the already published QNM frequencies and are found to coincide to a great extent with them. Also discussed are the QNM of the Kerr black hole, described by the Teukolsky Master equation.
21 pages, 3 figures, 4 tables; Amendments. Typos corrected. New sections and figures added. New comments on the application of the method in systems featuring confluent Heun functions, including the QNM of the Kerr black hole. Expanded numerical testing of the algorithm on simple systems; Internal Report, Sofia University, 2011
References in corpus (7)
- Quasinormal modes of black holes: from astrophysics to string theory
- Quasi-Normal Modes and Gravitational Wave Astronomy
- Heun Functions and Some of Their Applications in Physics
- Classes of Exact Solutions to the Teukolsky Master Equation
- Teukolsky-Starobinsky Identities - a Novel Derivation and Generalizations
- Black hole quasinormal modes: hints of quantum gravity?
- The Spectrum of Electromagnetic Jets from Kerr Black Holes and Naked Singularities in the Teukolsky Perturbation Theory